Cardinal number sentence example
- Two classes between which a one-one relation exists have the same cardinal number and are called cardinally similar; and the cardinal number of the class a is a certain class whose members are themselves classes - namely, it is the class composed of all those classes for which a one-one correlation with a exists.
- Thus the cardinal number of a is itself a class, and furthermore a is a member of it.
- Thus the cardinal number one is the class of unit classes, the cardinal number two is the class of doublets, and so on.
- The cardinal number zero is the class of classes with no members; but there is only one such class, namely - the null class.
- Zero is a cardinal number.Advertisement
- If a is a cardinal number, a+I is a cardinal number.
- If s is any class and zero is a member of it, also if when x is a cardinal number and a member of s, also x-}-I is a member of s, then the whole class of cardinal numbers is contained in s.
- The relation-number of a relation should be compared with the cardinal number of a class.
- Now if n be any finite cardinal number, it can be proved that the class of those serial relations, which have a field whose cardinal number is n, is a relation-number.
- The definition of the ordinal number requires some little ingenuity owing to the fact that no serial relation can have a field whose cardinal number is 1; but we must omit here the explanation of the process.Advertisement
- But the definition of the cardinal number of a class applies when the class is not finite, and it can be proved that there are different infinite cardinal numbers, and that there is a least infinite cardinal, now usually denoted by o where to is the Hebrew letter aleph.
- If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n X x = m X y.
- Thus the rational number one, which we will denote by ' r, is not the cardinal number I; for t r is the relation I/I as defined above, and is thus a relation holding between certain pairs of cardinals.
- In the above example 2 R is an integral real number, which is distinct from a rational integer, and from a cardinal number.
- Indeed, it is only by experience that we can know that any definite process of counting will give the true cardinal number of some class of entities.Advertisement
- Where a number is expressed in terms of various denominations, a cardinal number usually begins with the largest denomination, and an ordinal number with the smallest.